The Role of Independence and Stationarity in Probabilistic Models of Binary Choice

After more then 50 years of probabilistic choice modeling in economics, marketing, political science, psychology, and related disciplines, theoretical and computational advances give scholars access to a sophisticated array of modeling and inference resources. We review some important, but perhaps often overlooked, properties of major classes of probabilistic choice models. For within‐respondent applications, we discuss which models require repeated choices by an individual to be independent and response probabilities to be stationary. We show how some model classes, but not others, are invariant over variable preferences, variable utilities, or variable choice probabilities. These models, but not others, accommodate pooling of responses or averaging of choice proportions within participant when underlying parameters vary across observations. These, but not others, permit pooling/averaging across respondents in the presence of individual differences. We also review the role of independence and stationarity in statistical inference, including for probabilistic choice models that, themselves, do not require those properties. Copyright © 2017 John Wiley & Sons, Ltd.     

Emergence of spontaneous anticipatory hand movements in a probabilistic environment

In this article, we present a novel experimental approach to the study of anticipation in probabilistic cuing. We implemented a modified spatial cuing task in which participants made an anticipatory hand movement toward one of two probabilistic targets while the (x, y)-computer mouse coordinates of their hand movements were sampled. This approach allowed us to tap into anticipatory processes as they occurred, rather than just measuring their behavioral outcome through reaction time to the target. In different conditions, we varied the participants’ degree of certainty of the upcoming target position with probabilistic pre-cues. We found that participants initiated spontaneous anticipatory hand movements in all conditions, even when they had no information on the position of the upcoming target. However, participants’ hand position immediately before the target was affected by the degree of certainty concerning the target’s position. This modulation of anticipatory hand movements emerged rapidly in most participants as they encountered a constant probabilistic relation between a cue and an upcoming target position over the course of the experiment. Finally, we found individual differences in the way anticipatory behavior was modulated with an uncertain/neutral cue. Implications of these findings for probabilistic spatial cuing are discussed.     

Causal Inference from Descriptions of Experimental and Non-Experimental Research: Public Understanding of Correlation-Versus-Causation

The human tendency to conflate correlation with causation has been lamented by various scientists (Kida, 2006; Stanovich, 2009), and vivid examples of it can be found in both the media and peer-reviewed literature. However, there is little systematic data on the extent to which individuals conflate correlation with causation. In three experiments, we presented people with one of four research vignettes generated from the combination of two independent variables: whether the vignette described an experimental or non-experimental design, and whether it revealed a positive or negative association. Upon reading their vignette, participants selected inferences that could be drawn from the findings. Participants drew causal inferences from non-experimental vignettes as often as they did from experimental vignettes, and more frequently for causal statements and directions of association that fit with intuitive notions than for those that did not. We discuss our findings in relation to other biases in human thinking.     

程度函数及其可靠性

通过实验及理论分析,验证并给出了：１．程度函数ａ的可靠性θ的取值范围为：１／ｍ≤θ≤１／ｍ∑ｂｋｙｊｐ。θ值的大小与可靠性呈反向变化：ａ的可靠性大,θ值小,ａ的可靠性小θ值大,一般情况下的θ值,为θ的上界与下界之和的一半;２．采用集值统计模型和一般统计模型处理多级估量法多人次结果,所得到的程度函数平均数几乎相等,而且相关系数高达０．９７７．非常显著．分组数据计算、单一数据计算或全部数据统一计算的程度函数ａ及可靠性θ值也都相等．３．一般统计模型处理多级估量法多人次结果所得到的标准差σ,只反映各被试的程度函数之间的变异,与程度函数可靠性无关（相关系数０．００２４,非常不显著）．如果个人结果中按公式θ＝１／ｍ∑ｅｊ再计算可靠性θ值,然后再求其平均数及标准差,还可进一步了解可靠性θ值的变异情况．     

The repeated insertion model for rankings: Missing link between two subset choice models

Several probabilistic models for subset choice have been proposed in the literature, for example, to explain approval voting data. We show that Marley et al.'s latent scale model is subsumed by Falmagne and Regenwetter's size-independent model, in the sense that every choice probability distribution generated by the former can also be explained by the latter. Our proof relies on the construction of a probabilistic ranking model which we label the repeated insertion model. This model is a special case of Marden's orthogonal contrast model class and, in turn, includes the classical Mallows -model as a special case. We explore its basic properties as well as its relationship to Fligner and Verducci's multistage ranking model.The authors are grateful to the National Science Foundation for grants SES98-18756 to Regenwetter and Peke, and SBR97-30076 to Regenwetter. This collaborative research was carried out in the context of the conference Random Utility 2000 held at Duke University and sponsored by NSF, the Fuqua School of Business and the Center for International Business Education and Research. We thank the editor and four referees for helpful suggestions and we are grateful to Prof. J. I. Marden for providing useful information on contrast models. We thank Moon-Ho Ho for programming and running the data analyses.     

Contiguity and conditioned reinforcement in probabilistic choice

In a baseline condition, pigeons chose between an alternative that always provided food following a 30-s delay (100% reinforcement) and an alternative that provided food half of the time and blackout half of the time following 30-s delays (50% reinforcement). The different outcomes were signaled by different-colored keylights. On average, each alternative was chosen approximately equally often, replicating the finding of suboptimal choice in probabilistic reinforcement procedures. The efficacy of the delay stimuli (keylights) as conditioned reinforcers was assessed in other conditions by interposing a 5-s gap (keylights darkened) between the choice response and one or more of the delay stimuli. The strength of conditioned reinforcement was measured by the decrease in choice of an alternative when the alternative contained a gap. Preference for the 50% alternative decreased in conditions in which the gap preceded either all delay stimuli, both delay stimuli for the 50% alternative, or the food stimulus for the 50% alternative, but preference was not consistently affected in conditions in which the gap preceded only the 100% delay stimulus or the blackout stimulus for the 50% alternative. These results support the notion that conditioned reinforcement underlies the finding of suboptimal preference in probabilistic reinforcement procedures, and that the signal for food on the 50% reinforcement alternative functions as a stronger conditioned reinforcer than the signal for food on the 100% reinforcement alternative. In addition, the results fail to provide evidence that the signal for blackout functions as a conditioned punisher.     

Wide-Coverage Probabilistic Sentence Processing

This paper describes a fully implemented, broad-coverage model of human syntactic processing. The model uses probabilistic parsing techniques, which combine phrase structure, lexical category, and limited subcategory probabilities with an incremental, left-to-right pruning mechanism based on cascaded Markov models. The parameters of the system are established through a uniform training algorithm, which determines maximum-likelihood estimates from a parsed corpus. The probabilistic parsing mechanism enables the system to achieve good accuracy on typical, garden-variety language (i.e., when tested on corpora). Furthermore, the incremental probabilistic ranking of the preferred analyses during parsing also naturally explains observed human behavior for a range of garden-path structures. We do not make strong psychological claims about the specific probabilistic mechanism discussed here, which is limited by a number of practical considerations. Rather, we argue incremental probabilistic parsing models are, in general, extremely well suited to explaining this dual nature—generally good and occasionally pathological—of human linguistic performance.     

Age and probabilistic reasoning: Biases in conjunctive,disjunctive and Bayesian judgements in early and late adulthood

Probabilistic reasoning plays an essential part in many aspects of our daily routine and it has been argued that as we grow older, the need to make judgements under uncertainty becomes increasingly important. Two studies were conducted to establish whether the propensity to commit probabilistic reasoning errors increased with age. Young (aged 16–24), middle aged (25–54), and older persons (55 years and above) were included. Study 1 revealed systematic biases and errors across a range of judgement tasks. However, no evidence of any age effect in Bayesian inference, the incidence of the conjunction fallacy, or in the number of disjunction errors was found. The results obtained in Study 1 were replicated in Study 2, where the potential mediating role of working memory processes and intellectual capacity were explicitly assessed. While some aspects of probabilistic reasoning performance were correlated with measures of intelligence and working memory functioning among young adults, this was much less evident in older persons. The present findings are discussed in relation to the evolution of the dualistic heuristic–analytical system over the adult lifespan. Copyright © 2005 John Wiley & Sons, Ltd.